Question

hey I trying to study a test 4(x-2) -4 < or equal 3 (2+x) + 7 can u help me fully explain this lession

Answer 1

Sure! Let's solve the inequality step by step:

We want to solve the inequality:

\[ 4(x - 2) - 4 \leq 3(2 + x) + 7. \]

### Step 1: Expand both sides

**Left Side:**
1. Distribute the 4 into \( (x - 2) \):
\[
4(x - 2) = 4x - 8.
\]
2. Now combine with the -4:
\[
4x - 8 - 4 = 4x - 12.
\]

So, the left-hand side simplifies to \( 4x - 12 \).

**Right Side:**
1. Distribute the 3 into \( (2 + x) \):
\[
3(2 + x) = 6 + 3x.
\]
2. Now combine with the +7:
\[
6 + 3x + 7 = 3x + 13.
\]

So, the right-hand side simplifies to \( 3x + 13 \).

Now our inequality looks like this:

\[ 4x - 12 \leq 3x + 13. \]

### Step 2: Move the variables to one side

To isolate \( x \), let’s subtract \( 3x \) from both sides:

\[
4x - 3x - 12 \leq 13.
\]

This simplifies to:

\[
x - 12 \leq 13.
\]

### Step 3: Add 12 to both sides

Now, to isolate \( x \):

\[
x - 12 + 12 \leq 13 + 12.
\]

This simplifies to:

\[
x \leq 25.
\]

### Final Result:

So, the solution to the inequality \( 4(x - 2) - 4 \leq 3(2 + x) + 7 \) is:

\[
x \leq 25.
\]

### Interpretation:

This means that any value of \( x \) that is less than or equal to 25 will satisfy the original inequality.

### Conclusion:

Feel free to ask if you have any questions or if there's anything specific in this process you'd like to delve deeper into!